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plot(decompose(AirPassengers, type = "multiplicative"))

class(AirPassengers)

[1] "ts"

plot(AirPassengers)

beerprod <- scan("/Users/ansonleung/Desktop/Applied Financial Engineering/Financial Times Series Analysis/Lecture 1/R Code/beerprod.dat")

Read 72 items

beerprod = ts(beerprod, freq = 4)
plot(stl(beerprod, "periodic"))

trendpattern = filter(beerprod, filter = c(1/8, 1/4, 1/4, 1/4, 1/8), sides=2)
plot(beerprod, type= "b", main = "moving average annual trend")
lines(trendpattern)

seasonals = beerprod - trendpattern
plot(seasonals, type = "b", main = "Seasonal pattern for beer production")

trendpattern2 = filter(beerprod, filter = c(1/4, 1/4, 1/4, 1/4), sides=1)
plot(beerprod, type= "b", main = "moving average annual trend")
lines(trendpattern2)

options(digits = 9)
data(bmw, package = "evir")
Box.test(bmw, lag = 5, type = "Ljung-Box")

    Box-Ljung test

data:  bmw
X-squared = 44.98701, df = 5, p-value = 1.45972e-08

arima(x=bmw, order=c(1,0,0))

Call:
arima(x = bmw, order = c(1, 0, 0))

Coefficients:
           ar1  intercept
      0.081116   0.000340
s.e.  0.012722   0.000205

sigma^2 estimated as 0.00021626:  log likelihood = 17212.34,  aic = -34418.68

fitAR1 = arima(bmw, order = c(1,0,0))
Box.test(residuals(fitAR1), lag = 10, type = "Ljung-Box", fitdf = 1)

    Box-Ljung test

data:  residuals(fitAR1)
X-squared = 17.51319, df = 9, p-value = 0.0412603

Box.test(residuals(fitAR1), lag = 15, type = "Ljung-Box", fitdf = 1)

    Box-Ljung test

data:  residuals(fitAR1)
X-squared = 24.09947, df = 14, p-value = 0.0445707

Box.test(residuals(fitAR1), lag = 20, type = "Ljung-Box", fitdf = 1)

    Box-Ljung test

data:  residuals(fitAR1)
X-squared = 39.29885, df = 19, p-value = 0.00404028

data(Mishkin, package = "Ecdat")

x = as.ts(Mishkin[,1], start=1950, frequency=12)
acf(x)

pacf(x)

auto.arima(diff(x), max.p=20, max.q=0,ic="aic")

Series: diff(x) 
ARIMA(10,0,0)(2,0,0)[12] with zero mean 

Coefficients:
            ar1        ar2        ar3        ar4        ar5        ar6        ar7        ar8
      -0.639976  -0.522701  -0.557229  -0.468459  -0.404792  -0.326150  -0.221002  -0.175748
s.e.   0.045994   0.055234   0.060157   0.065184   0.067415   0.067535   0.066081   0.060262
            ar9       ar10      sar1       sar2
      -0.087721  -0.069657  0.089960  -0.032370
s.e.   0.055756   0.047502  0.050207   0.051833

sigma^2 estimated as 8.61379:  log likelihood=-1217.38
AIC=2460.75   AICc=2461.52   BIC=2515.28

auto.arima(diff(x), max.p=10, max.q=0,ic="bic")

Series: diff(x) 
ARIMA(6,0,0)(2,0,0)[12] with zero mean 

Coefficients:
            ar1        ar2        ar3        ar4        ar5        ar6      sar1       sar2
      -0.608313  -0.459024  -0.470350  -0.352204  -0.260538  -0.160104  0.101906  -0.010042
s.e.   0.045463   0.052406   0.054742   0.055632   0.053044   0.046181  0.048743   0.050094

sigma^2 estimated as 8.7618:  log likelihood=-1223.49
AIC=2464.98   AICc=2465.36   BIC=2502.73

data(Mishkin, package = "Ecdat")

y = as.vector(Mishkin[,1])
fitMA3 = arima(diff(y), order = c(0,0,3))
fitMA3 

Call:
arima(x = diff(y), order = c(0, 0, 3))

Coefficients:
            ma1        ma2        ma3  intercept
      -0.632950  -0.102734  -0.108172  -0.000156
s.e.   0.046017   0.051399   0.046985   0.020892

sigma^2 estimated as 8.5046:  log likelihood = -1220.26,  aic = 2450.52

Box.test(fitMA3$residuals, lag = 5, type = "Ljung", fitdf = 3)

    Box-Ljung test

data:  fitMA3$residuals
X-squared = 0.862017, df = 2, p-value = 0.649853

Box.test(fitMA3$residuals, lag = 10, type = "Ljung", fitdf = 3)

    Box-Ljung test

data:  fitMA3$residuals
X-squared = 4.205325, df = 7, p-value = 0.755848

Box.test(fitMA3$residuals, lag = 15, type = "Ljung", fitdf = 3)

    Box-Ljung test

data:  fitMA3$residuals
X-squared = 13.71089, df = 12, p-value = 0.31955

Box.test(fitMA3$residuals, lag = 20, type = "Ljung", fitdf = 3)

    Box-Ljung test

data:  fitMA3$residuals
X-squared = 26.45871, df = 17, p-value = 0.0664989

fitMA2 = arima(diff(y), order = c(0,0,2))
fitMA2 

Call:
arima(x = diff(y), order = c(0, 0, 2))

Coefficients:
            ma1        ma2  intercept
      -0.662088  -0.166016   0.000145
s.e.   0.041879   0.040721   0.023044

sigma^2 estimated as 8.5951:  log likelihood = -1222.85,  aic = 2453.7

Box.test(fitMA2$residuals, lag = 5, type = "Ljung", fitdf = 2)

    Box-Ljung test

data:  fitMA2$residuals
X-squared = 6.608554, df = 3, p-value = 0.0854783

Box.test(fitMA2$residuals, lag = 10, type = "Ljung", fitdf = 2)

    Box-Ljung test

data:  fitMA2$residuals
X-squared = 8.838851, df = 8, p-value = 0.356072

Box.test(fitMA2$residuals, lag = 15, type = "Ljung", fitdf = 2)

    Box-Ljung test

data:  fitMA2$residuals
X-squared = 15.8254, df = 13, p-value = 0.258687

Box.test(fitMA2$residuals, lag = 20, type = "Ljung", fitdf = 2)

    Box-Ljung test

data:  fitMA2$residuals
X-squared = 29.48198, df = 18, p-value = 0.0427975

data(Capm, package = "Ecdat")
y = as.ts(Capm[,1], start=1960-01, frequency=12)

fitARMA1 = arima(y, order = c(1,0,1))
fitARMA1

Call:
arima(x = y, order = c(1, 0, 1))

Coefficients:
            ar1       ma1  intercept
      -0.818604  0.896077   0.665177
s.e.   0.088075  0.068410   0.206508

sigma^2 estimated as 20.2452:  log likelihood = -1508.26,  aic = 3024.52

acf(residuals(fitARMA1),lag.max=20,main="ARMA1")

qqnorm(residuals(fitARMA1),main="ARMA1") ; qqline(residuals(fitARMA1))

plot(residuals(fitARMA1))

polyroot(c(1,-1.229,+0.233))

[1] 1.00525088+0i 4.26942723-0i

data(Mishkin, package = 'Ecdat')
y = as.vector(Mishkin[,1])

adf.test(y)

    Augmented Dickey-Fuller Test

data:  y
Dickey-Fuller = -3.865109, Lag order = 7, p-value = 0.0157645
alternative hypothesis: stationary

pp.test(y)

p-value smaller than printed p-value

    Phillips-Perron Unit Root Test

data:  y
Dickey-Fuller Z(alpha) = -248.7537, Truncation lag parameter = 5, p-value = 0.01
alternative hypothesis: stationary

kpss.test(y)

p-value smaller than printed p-value

    KPSS Test for Level Stationarity

data:  y
KPSS Level = 2.510004, Truncation lag parameter = 5, p-value = 0.01

auto.arima(y, max.p=5, max.q = 5, ic='aic', trace = FALSE)

Series: y 
ARIMA(1,1,1) 

Coefficients:
           ar1        ma1
      0.238309  -0.877177
s.e.  0.055043   0.026929

sigma^2 estimated as 8.58719:  log likelihood=-1221.62
AIC=2449.25   AICc=2449.29   BIC=2461.83

auto.arima(y, max.p=5, max.q = 5, ic='bic', trace = FALSE)

Series: y 
ARIMA(1,1,1) 

Coefficients:
           ar1        ma1
      0.238309  -0.877177
s.e.  0.055043   0.026929

sigma^2 estimated as 8.58719:  log likelihood=-1221.62
AIC=2449.25   AICc=2449.29   BIC=2461.83

fitARIMA111= arima(y, c(1,1,1))
par(mfrow=c(1,1))
acf(fitARIMA111$resid)

Box.test(fitARIMA111$resid, lag = 15, fitdf =2)

    Box-Pierce test

data:  fitARIMA111$resid
X-squared = 14.44181, df = 13, p-value = 0.343481

data(Hstarts, package="Ecdat") 
x = ts(Hstarts[,1], start=1960, frequency=4)

fit1 = arima(x, c(1,1,1), seasonal = list(order = c(1,1,1), period = 4))
fit1

Call:
arima(x = x, order = c(1, 1, 1), seasonal = list(order = c(1, 1, 1), period = 4))

Coefficients:
           ar1        ma1       sar1       sma1
      0.688314  -0.881866  -0.155035  -0.765925
s.e.  0.238596   0.187552   0.097431   0.081197

sigma^2 estimated as 0.0257251:  log likelihood = 64.13,  aic = -118.27

fit2 = arima(x, c(1,1,1), seasonal = list(order = c(0,1,1), period = 4))
fit2

Call:
arima(x = x, order = c(1, 1, 1), seasonal = list(order = c(0, 1, 1), period = 4))

Coefficients:
           ar1        ma1       sma1
      0.674779  -0.890084  -0.822011
s.e.  0.141937   0.104565   0.051223

sigma^2 estimated as 0.0260917:  log likelihood = 62.91,  aic = -117.82

Box.test(residuals(fit2), type="Ljung-Box", lag = 10, fitdf=3)

    Box-Ljung test

data:  residuals(fit2)
X-squared = 8.449959, df = 7, p-value = 0.294592

Box.test(residuals(fit1), type="Ljung-Box", lag = 10, fitdf=4)

    Box-Ljung test

data:  residuals(fit1)
X-squared = 5.821895, df = 6, p-value = 0.443435

library("fracdiff")
library("longmemo")

D = c(-.35, .35)

set.seed("09201948")

par(mfrow=c(length(D)+1,2))

for (i in 1:length(D))
{
H = D[i] + 1/2
x = simARMA0(2500,H)
plot(x,main=toString(paste("d =", D[i])) )
acf(x,main=toString(paste("d =", D[i])) )
}

d= .7-1
H = d + 1/2
y = simARMA0(2500,H)
x = cumsum(y)
plot(x,main="d = 0.7",type="l")

acf(x,main="d = 0.7")

par(mfrow=c(1,2))
acf(diffseries(x,.7),main =expression(paste(Delta^0.7, Y)))
acf(diff(x),main = expression(paste(Delta, Y)))

n = 10500
set.seed("2340")
e = rnorm(n)
a = e
y = e
sig2= 2^2
omega = 1
alpha =  0.08
beta = 0.90
phi = 0.8
mu = 0.1

for (t in 2:n)
{
a[t] = sqrt(sig2[t])*e[t]
y[t] = mu + phi*(y[t-1]-mu) + a[t]
sig2[t+1] = omega + alpha*a[t]^2 + beta*sig2[t]
}

pdf ("garcho3.pdf", width = 9, height = 6)
#

par(mfrow=c(2,4))

plot(e[10001:n], type="l",xlab="t", ylab = expression(epsilon), main="(a) white noise")

plot(sqrt(sig2[10001:n]), type="l",xlab="t",ylim=expression(sigma[t]), main="(b) conditional std dev")

Error in plot.window(...) : invalid 'ylim' value

library(xts)

Loading required package: zoo

Attaching package: ‘zoo’

The following objects are masked from ‘package:base’:

    as.Date, as.Date.numeric

library(fGarch)

Loading required package: timeDate
Loading required package: timeSeries

Attaching package: ‘timeSeries’

The following object is masked from ‘package:zoo’:

    time<-

Loading required package: fBasics

garchFit(formula = ~arma(1,0) + garch(1, 1), data = bmw, cond.dist = "norm")

Series Initialization:
 ARMA Model:                arma
 Formula Mean:              ~ arma(1, 0)
 GARCH Model:               garch
 Formula Variance:          ~ garch(1, 1)
 ARMA Order:                1 0
 Max ARMA Order:            1
 GARCH Order:               1 1
 Max GARCH Order:           1
 Maximum Order:             1
 Conditional Dist:          norm
 h.start:                   2
 llh.start:                 1
 Length of Series:          6146
 Recursion Init:            mci
 Series Scale:              0.0147555259

Parameter Initialization:
 Initial Parameters:          $params
 Limits of Transformations:   $U, $V
 Which Parameters are Fixed?  $includes
 Parameter Matrix:

 Index List of Parameters to be Optimized:
    mu    ar1  omega alpha1  beta1 
     1      2      3      4      6 
 Persistence:                  0.9 


--- START OF TRACE ---
Selected Algorithm: nlminb 

R coded nlminb Solver: 

  0:     8197.7076: 0.0231375 0.0811175 0.100000 0.100000 0.800000
  1:     8191.4368: 0.0231376 0.0820231 0.0829234 0.101366 0.792310
  2:     8171.2956: 0.0231377 0.0834840 0.0819964 0.118120 0.800662
  3:     8165.8147: 0.0231379 0.0858254 0.0654065 0.126643 0.800937
  4:     8161.3630: 0.0231384 0.0913135 0.0611973 0.134195 0.816703
  5:     8157.6333: 0.0231397 0.0999004 0.0505134 0.126605 0.827092
  6:     8156.0110: 0.0231426 0.104737 0.0482985 0.115063 0.840945
  7:     8155.6452: 0.0231478 0.0932725 0.0422975 0.107529 0.852311
  8:     8155.5196: 0.0231478 0.0933535 0.0431822 0.107833 0.852922
  9:     8155.4436: 0.0231494 0.0943324 0.0430573 0.107351 0.852714
 10:     8155.4195: 0.0231526 0.0961691 0.0435774 0.106424 0.853420
 11:     8155.3582: 0.0231639 0.0995904 0.0416707 0.105718 0.855419
 12:     8155.3059: 0.0232332 0.0979839 0.0420297 0.103245 0.856914
 13:     8155.3026: 0.0232332 0.0979930 0.0420814 0.103311 0.856994
 14:     8155.3008: 0.0232333 0.0980164 0.0419790 0.103351 0.857025
 15:     8155.2995: 0.0232358 0.0980415 0.0419474 0.103372 0.857226
 16:     8155.2958: 0.0232430 0.0980644 0.0417613 0.103189 0.857457
 17:     8155.2923: 0.0232594 0.0981493 0.0415880 0.102942 0.857970
 18:     8155.2896: 0.0233012 0.0984606 0.0414384 0.102753 0.858151
 19:     8155.2873: 0.0233442 0.0985178 0.0414683 0.102771 0.858211
 20:     8155.2833: 0.0234833 0.0978777 0.0412894 0.102594 0.858545
 21:     8155.2571: 0.0249755 0.0992718 0.0413171 0.103253 0.857706
 22:     8155.2462: 0.0256650 0.100021 0.0408549 0.101828 0.860009
 23:     8155.2300: 0.0263554 0.0991767 0.0414259 0.103072 0.858170
 24:     8155.2279: 0.0263554 0.0991727 0.0413643 0.103032 0.858128
 25:     8155.2271: 0.0263556 0.0990221 0.0414014 0.102795 0.858300
 26:     8155.2266: 0.0263708 0.0990310 0.0413480 0.102755 0.858330
 27:     8155.2238: 0.0266534 0.0992512 0.0410613 0.102259 0.859106
 28:     8155.2220: 0.0269367 0.0989338 0.0410123 0.102234 0.859194
 29:     8155.2214: 0.0271533 0.0985704 0.0408774 0.102070 0.859482
 30:     8155.2214: 0.0271811 0.0986000 0.0409012 0.102100 0.859430
 31:     8155.2214: 0.0271709 0.0985960 0.0408969 0.102096 0.859439

Final Estimate of the Negative LLH:
 LLH:  -17757.1604    norm LLH:  -2.88922233 
            mu            ar1          omega         alpha1          beta1 
4.00921646e-04 9.85960084e-02 8.90429955e-06 1.02096121e-01 8.59438881e-01 

R-optimhess Difference Approximated Hessian Matrix:
                    mu             ar1           omega          alpha1           beta1
mu     -4.03227080e+07    -9110.700614 -2.47133587e+08 -1.02365122e+04  1.56766704e+02
ar1    -9.11070061e+03    -4921.397928 -3.17579246e+06 -3.30680113e+02 -8.62839816e+02
omega  -2.47133587e+08 -3175792.459279 -7.19620950e+12 -7.10411367e+08 -1.06585562e+09
alpha1 -1.02365122e+04     -330.680113 -7.10411367e+08 -1.15660301e+05 -1.34997175e+05
beta1   1.56766704e+02     -862.839816 -1.06585562e+09 -1.34997175e+05 -1.81389074e+05
attr(,"time")
Time difference of 0.0703678131 secs

--- END OF TRACE ---


Time to Estimate Parameters:
 Time difference of 0.372771025 secs

Using formula(x) is deprecated when x is a character vector of length > 1.
  Consider formula(paste(x, collapse = " ")) instead.

Title:
 GARCH Modelling 

Call:
 garchFit(formula = ~arma(1, 0) + garch(1, 1), data = bmw, cond.dist = "norm") 

Mean and Variance Equation:
 data ~ arma(1, 0) + garch(1, 1)
<environment: 0x7fe8b7508820>
 [data = bmw]

Conditional Distribution:
 norm 

Coefficient(s):
        mu         ar1       omega      alpha1       beta1  
4.0092e-04  9.8596e-02  8.9043e-06  1.0210e-01  8.5944e-01  

Std. Errors:
 based on Hessian 

Error Analysis:
        Estimate  Std. Error  t value Pr(>|t|)    
mu     4.009e-04   1.579e-04    2.539   0.0111 *  
ar1    9.860e-02   1.431e-02    6.888 5.65e-12 ***
omega  8.904e-06   1.449e-06    6.145 7.97e-10 ***
alpha1 1.021e-01   1.135e-02    8.994  < 2e-16 ***
beta1  8.594e-01   1.581e-02   54.348  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Log Likelihood:
 17757.1604    normalized:  2.88922233 

Description:
 Sun Aug  1 10:17:33 2021 by user:  

summary(garchFit(formula = ~arma(1, 0) + garch(1, 1), data = bmw, cond.dist = "norm"))

Series Initialization:
 ARMA Model:                arma
 Formula Mean:              ~ arma(1, 0)
 GARCH Model:               garch
 Formula Variance:          ~ garch(1, 1)
 ARMA Order:                1 0
 Max ARMA Order:            1
 GARCH Order:               1 1
 Max GARCH Order:           1
 Maximum Order:             1
 Conditional Dist:          norm
 h.start:                   2
 llh.start:                 1
 Length of Series:          6146
 Recursion Init:            mci
 Series Scale:              0.0147555259

Parameter Initialization:
 Initial Parameters:          $params
 Limits of Transformations:   $U, $V
 Which Parameters are Fixed?  $includes
 Parameter Matrix:

 Index List of Parameters to be Optimized:
    mu    ar1  omega alpha1  beta1 
     1      2      3      4      6 
 Persistence:                  0.9 


--- START OF TRACE ---
Selected Algorithm: nlminb 

R coded nlminb Solver: 

  0:     8197.7076: 0.0231375 0.0811175 0.100000 0.100000 0.800000
  1:     8191.4368: 0.0231376 0.0820231 0.0829234 0.101366 0.792310
  2:     8171.2956: 0.0231377 0.0834840 0.0819964 0.118120 0.800662
  3:     8165.8147: 0.0231379 0.0858254 0.0654065 0.126643 0.800937
  4:     8161.3630: 0.0231384 0.0913135 0.0611973 0.134195 0.816703
  5:     8157.6333: 0.0231397 0.0999004 0.0505134 0.126605 0.827092
  6:     8156.0110: 0.0231426 0.104737 0.0482985 0.115063 0.840945
  7:     8155.6452: 0.0231478 0.0932725 0.0422975 0.107529 0.852311
  8:     8155.5196: 0.0231478 0.0933535 0.0431822 0.107833 0.852922
  9:     8155.4436: 0.0231494 0.0943324 0.0430573 0.107351 0.852714
 10:     8155.4195: 0.0231526 0.0961691 0.0435774 0.106424 0.853420
 11:     8155.3582: 0.0231639 0.0995904 0.0416707 0.105718 0.855419
 12:     8155.3059: 0.0232332 0.0979839 0.0420297 0.103245 0.856914
 13:     8155.3026: 0.0232332 0.0979930 0.0420814 0.103311 0.856994
 14:     8155.3008: 0.0232333 0.0980164 0.0419790 0.103351 0.857025
 15:     8155.2995: 0.0232358 0.0980415 0.0419474 0.103372 0.857226
 16:     8155.2958: 0.0232430 0.0980644 0.0417613 0.103189 0.857457
 17:     8155.2923: 0.0232594 0.0981493 0.0415880 0.102942 0.857970
 18:     8155.2896: 0.0233012 0.0984606 0.0414384 0.102753 0.858151
 19:     8155.2873: 0.0233442 0.0985178 0.0414683 0.102771 0.858211
 20:     8155.2833: 0.0234833 0.0978777 0.0412894 0.102594 0.858545
 21:     8155.2571: 0.0249755 0.0992718 0.0413171 0.103253 0.857706
 22:     8155.2462: 0.0256650 0.100021 0.0408549 0.101828 0.860009
 23:     8155.2300: 0.0263554 0.0991767 0.0414259 0.103072 0.858170
 24:     8155.2279: 0.0263554 0.0991727 0.0413643 0.103032 0.858128
 25:     8155.2271: 0.0263556 0.0990221 0.0414014 0.102795 0.858300
 26:     8155.2266: 0.0263708 0.0990310 0.0413480 0.102755 0.858330
 27:     8155.2238: 0.0266534 0.0992512 0.0410613 0.102259 0.859106
 28:     8155.2220: 0.0269367 0.0989338 0.0410123 0.102234 0.859194
 29:     8155.2214: 0.0271533 0.0985704 0.0408774 0.102070 0.859482
 30:     8155.2214: 0.0271811 0.0986000 0.0409012 0.102100 0.859430
 31:     8155.2214: 0.0271709 0.0985960 0.0408969 0.102096 0.859439

Final Estimate of the Negative LLH:
 LLH:  -17757.1604    norm LLH:  -2.88922233 
            mu            ar1          omega         alpha1          beta1 
4.00921646e-04 9.85960084e-02 8.90429955e-06 1.02096121e-01 8.59438881e-01 

R-optimhess Difference Approximated Hessian Matrix:
                    mu             ar1           omega          alpha1           beta1
mu     -4.03227080e+07    -9110.700614 -2.47133587e+08 -1.02365122e+04  1.56766704e+02
ar1    -9.11070061e+03    -4921.397928 -3.17579246e+06 -3.30680113e+02 -8.62839816e+02
omega  -2.47133587e+08 -3175792.459279 -7.19620950e+12 -7.10411367e+08 -1.06585562e+09
alpha1 -1.02365122e+04     -330.680113 -7.10411367e+08 -1.15660301e+05 -1.34997175e+05
beta1   1.56766704e+02     -862.839816 -1.06585562e+09 -1.34997175e+05 -1.81389074e+05
attr(,"time")
Time difference of 0.067045927 secs

--- END OF TRACE ---


Time to Estimate Parameters:
 Time difference of 0.349786997 secs

Using formula(x) is deprecated when x is a character vector of length > 1.
  Consider formula(paste(x, collapse = " ")) instead.

Title:
 GARCH Modelling 

Call:
 garchFit(formula = ~arma(1, 0) + garch(1, 1), data = bmw, cond.dist = "norm") 

Mean and Variance Equation:
 data ~ arma(1, 0) + garch(1, 1)
<environment: 0x7fe8d64ecea0>
 [data = bmw]

Conditional Distribution:
 norm 

Coefficient(s):
        mu         ar1       omega      alpha1       beta1  
4.0092e-04  9.8596e-02  8.9043e-06  1.0210e-01  8.5944e-01  

Std. Errors:
 based on Hessian 

Error Analysis:
        Estimate  Std. Error  t value Pr(>|t|)    
mu     4.009e-04   1.579e-04    2.539   0.0111 *  
ar1    9.860e-02   1.431e-02    6.888 5.65e-12 ***
omega  8.904e-06   1.449e-06    6.145 7.97e-10 ***
alpha1 1.021e-01   1.135e-02    8.994  < 2e-16 ***
beta1  8.594e-01   1.581e-02   54.348  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Log Likelihood:
 17757.1604    normalized:  2.88922233 

Description:
 Sun Aug  1 10:17:34 2021 by user:  


Standardised Residuals Tests:
                                Statistic  p-Value     
 Jarque-Bera Test   R    Chi^2  11377.9945 0           
 Shapiro-Wilk Test  R    W      NA         NA          
 Ljung-Box Test     R    Q(10)  15.1569254 0.126444518 
 Ljung-Box Test     R    Q(15)  20.0934511 0.168377031 
 Ljung-Box Test     R    Q(20)  30.5478816 0.0614491696
 Ljung-Box Test     R^2  Q(10)  5.03271737 0.88898177  
 Ljung-Box Test     R^2  Q(15)  7.53927234 0.940924545 
 Ljung-Box Test     R^2  Q(20)  9.27722902 0.979468118 
 LM Arch Test       R    TR^2   6.0325397  0.914432863 

Information Criterion Statistics:
        AIC         BIC         SIC        HQIC 
-5.77681758 -5.77134772 -5.77681890 -5.77492037 

library(xts)
library(rugarch)

Loading required package: parallel

Attaching package: ‘rugarch’

The following object is masked from ‘package:stats’:

    sigma

data(bmw, package="evir")
arma.garch.norm = ugarchspec(mean.model=list(armaOrder=c(1,0)), variance.model=list(garchOrder=c(1,1)))
bmw.garch.norm = ugarchfit(data=bmw, spec=arma.garch.norm)
show(bmw.garch.norm)

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(1,0,0)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.000453    0.000175   2.5937 0.009495
ar1     0.098133    0.014261   6.8811 0.000000
omega   0.000009    0.000000  23.0627 0.000000
alpha1  0.099405    0.005593  17.7730 0.000000
beta1   0.863666    0.006283 137.4528 0.000000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.000453    0.000186   2.4380  0.01477
ar1     0.098133    0.014073   6.9732  0.00000
omega   0.000009    0.000001  13.4483  0.00000
alpha1  0.099405    0.007903  12.5785  0.00000
beta1   0.863666    0.010529  82.0234  0.00000

LogLikelihood : 17751.9289 

Information Criteria
------------------------------------
                    
Akaike       -5.7751
Bayes        -5.7696
Shibata      -5.7751
Hannan-Quinn -5.7732

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.7790  0.3775
Lag[2*(p+q)+(p+q)-1][2]    0.9161  0.7890
Lag[4*(p+q)+(p+q)-1][5]    3.3274  0.3536
d.o.f=1
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.2769  0.5988
Lag[2*(p+q)+(p+q)-1][5]    1.0262  0.8537
Lag[4*(p+q)+(p+q)-1][9]    1.7209  0.9356
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]    0.1922 0.500 2.000  0.6611
ARCH Lag[5]    1.1094 1.440 1.667  0.7008
ARCH Lag[7]    1.2290 2.315 1.543  0.8737

Nyblom stability test
------------------------------------
Joint Statistic:  38.6063
Individual Statistics:              
mu     0.05867
ar1    0.21574
omega  5.66190
alpha1 0.18834
beta1  0.18245

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.28 1.47 1.88
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------

Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     493.2    1.477e-92
2    30     513.4    4.975e-90
3    40     559.3    2.484e-93
4    50     585.5    5.611e-93


Elapsed time : 0.687120914 

plot(bmw.garch.norm, which="all")

please wait...calculating quantiles...

par(mfrow = c(3,2))

for(i in c(1,3,10,11,8,9)) 
  plot(bmw.garch.norm, which=i)

library(MASS)
bmw.garch.norm <- garchFit(formula = ~arma(1, 0) + garch(1, 1), data = bmw, cond.dist = "norm")

Series Initialization:
 ARMA Model:                arma
 Formula Mean:              ~ arma(1, 0)
 GARCH Model:               garch
 Formula Variance:          ~ garch(1, 1)
 ARMA Order:                1 0
 Max ARMA Order:            1
 GARCH Order:               1 1
 Max GARCH Order:           1
 Maximum Order:             1
 Conditional Dist:          norm
 h.start:                   2
 llh.start:                 1
 Length of Series:          6146
 Recursion Init:            mci
 Series Scale:              0.0147555259

Parameter Initialization:
 Initial Parameters:          $params
 Limits of Transformations:   $U, $V
 Which Parameters are Fixed?  $includes
 Parameter Matrix:

 Index List of Parameters to be Optimized:
    mu    ar1  omega alpha1  beta1 
     1      2      3      4      6 
 Persistence:                  0.9 


--- START OF TRACE ---
Selected Algorithm: nlminb 

R coded nlminb Solver: 

  0:     8197.7076: 0.0231375 0.0811175 0.100000 0.100000 0.800000
  1:     8191.4368: 0.0231376 0.0820231 0.0829234 0.101366 0.792310
  2:     8171.2956: 0.0231377 0.0834840 0.0819964 0.118120 0.800662
  3:     8165.8147: 0.0231379 0.0858254 0.0654065 0.126643 0.800937
  4:     8161.3630: 0.0231384 0.0913135 0.0611973 0.134195 0.816703
  5:     8157.6333: 0.0231397 0.0999004 0.0505134 0.126605 0.827092
  6:     8156.0110: 0.0231426 0.104737 0.0482985 0.115063 0.840945
  7:     8155.6452: 0.0231478 0.0932725 0.0422975 0.107529 0.852311
  8:     8155.5196: 0.0231478 0.0933535 0.0431822 0.107833 0.852922
  9:     8155.4436: 0.0231494 0.0943324 0.0430573 0.107351 0.852714
 10:     8155.4195: 0.0231526 0.0961691 0.0435774 0.106424 0.853420
 11:     8155.3582: 0.0231639 0.0995904 0.0416707 0.105718 0.855419
 12:     8155.3059: 0.0232332 0.0979839 0.0420297 0.103245 0.856914
 13:     8155.3026: 0.0232332 0.0979930 0.0420814 0.103311 0.856994
 14:     8155.3008: 0.0232333 0.0980164 0.0419790 0.103351 0.857025
 15:     8155.2995: 0.0232358 0.0980415 0.0419474 0.103372 0.857226
 16:     8155.2958: 0.0232430 0.0980644 0.0417613 0.103189 0.857457
 17:     8155.2923: 0.0232594 0.0981493 0.0415880 0.102942 0.857970
 18:     8155.2896: 0.0233012 0.0984606 0.0414384 0.102753 0.858151
 19:     8155.2873: 0.0233442 0.0985178 0.0414683 0.102771 0.858211
 20:     8155.2833: 0.0234833 0.0978777 0.0412894 0.102594 0.858545
 21:     8155.2571: 0.0249755 0.0992718 0.0413171 0.103253 0.857706
 22:     8155.2462: 0.0256650 0.100021 0.0408549 0.101828 0.860009
 23:     8155.2300: 0.0263554 0.0991767 0.0414259 0.103072 0.858170
 24:     8155.2279: 0.0263554 0.0991727 0.0413643 0.103032 0.858128
 25:     8155.2271: 0.0263556 0.0990221 0.0414014 0.102795 0.858300
 26:     8155.2266: 0.0263708 0.0990310 0.0413480 0.102755 0.858330
 27:     8155.2238: 0.0266534 0.0992512 0.0410613 0.102259 0.859106
 28:     8155.2220: 0.0269367 0.0989338 0.0410123 0.102234 0.859194
 29:     8155.2214: 0.0271533 0.0985704 0.0408774 0.102070 0.859482
 30:     8155.2214: 0.0271811 0.0986000 0.0409012 0.102100 0.859430
 31:     8155.2214: 0.0271709 0.0985960 0.0408969 0.102096 0.859439

Final Estimate of the Negative LLH:
 LLH:  -17757.1604    norm LLH:  -2.88922233 
            mu            ar1          omega         alpha1          beta1 
4.00921646e-04 9.85960084e-02 8.90429955e-06 1.02096121e-01 8.59438881e-01 

R-optimhess Difference Approximated Hessian Matrix:
                    mu             ar1           omega          alpha1           beta1
mu     -4.03227080e+07    -9110.700614 -2.47133587e+08 -1.02365122e+04  1.56766704e+02
ar1    -9.11070061e+03    -4921.397928 -3.17579246e+06 -3.30680113e+02 -8.62839816e+02
omega  -2.47133587e+08 -3175792.459279 -7.19620950e+12 -7.10411367e+08 -1.06585562e+09
alpha1 -1.02365122e+04     -330.680113 -7.10411367e+08 -1.15660301e+05 -1.34997175e+05
beta1   1.56766704e+02     -862.839816 -1.06585562e+09 -1.34997175e+05 -1.81389074e+05
attr(,"time")
Time difference of 0.065267086 secs

--- END OF TRACE ---


Time to Estimate Parameters:
 Time difference of 0.342492104 secs

Using formula(x) is deprecated when x is a character vector of length > 1.
  Consider formula(paste(x, collapse = " ")) instead.

e = residuals(bmw.garch.norm, standardize=TRUE)
fitdistr(e,"t")

NaNs producedNaNs producedNaNs producedNaNs producedNaNs produced

        m               s               df      
  -0.0239865051    0.7263501855    4.1097516314 
 ( 0.0109243437) ( 0.0121327797) ( 0.2357963744)

n = length(e)
grid = (1:n)/(n+1)
qqplot(sort(as.numeric(e)), qt(grid,df=4), main="t-plot, df=4",xlab= "Standardized residual quantiles",ylab="t-quantiles")

qqnorm(sort(as.numeric(e)));
qqline(y, col = 2)

garchFit(formula = ~arma(1, 1) + garch(1, 1), data = bmw,
cond.dist = "std")

Series Initialization:
 ARMA Model:                arma
 Formula Mean:              ~ arma(1, 1)
 GARCH Model:               garch
 Formula Variance:          ~ garch(1, 1)
 ARMA Order:                1 1
 Max ARMA Order:            1
 GARCH Order:               1 1
 Max GARCH Order:           1
 Maximum Order:             1
 Conditional Dist:          std
 h.start:                   2
 llh.start:                 1
 Length of Series:          6146
 Recursion Init:            mci
 Series Scale:              0.0147555259

Parameter Initialization:
 Initial Parameters:          $params
 Limits of Transformations:   $U, $V
 Which Parameters are Fixed?  $includes
 Parameter Matrix:

 Index List of Parameters to be Optimized:
    mu    ar1    ma1  omega alpha1  beta1  shape 
     1      2      3      4      5      7     10 
 Persistence:                  0.9 


--- START OF TRACE ---
Selected Algorithm: nlminb 

R coded nlminb Solver: 

  0:     7788.2143: 0.0231079 -0.247788 0.331458 0.100000 0.100000 0.800000  4.00000
  1:     7780.8363: 0.0231063 -0.250241 0.329122 0.0726877 0.115229 0.795726  3.99979
  2:     7768.7866: 0.0231043 -0.252470 0.327035 0.0722958 0.142456 0.811750  4.00011
  3:     7764.5716: 0.0231000 -0.254259 0.325523 0.0449827 0.158395 0.813199  3.99993
  4:     7758.0833: 0.0230808 -0.251634 0.329272 0.0459098 0.136742 0.835893  3.99879
  5:     7756.4790: 0.0230660 -0.259996 0.321726 0.0413723 0.112864 0.852912  3.99854
  6:     7755.9517: 0.0230500 -0.259088 0.323686 0.0312476 0.103218 0.881320  3.99858
  7:     7755.2243: 0.0230443 -0.255528 0.327670 0.0250334 0.0966100 0.884447  3.99868
  8:     7754.1836: 0.0230340 -0.251103 0.333020 0.0318698 0.0980444 0.879602  3.99876
  9:     7753.8694: 0.0230211 -0.258499 0.326588 0.0319674 0.101967 0.876632  3.99903
 10:     7753.8184: 0.0230210 -0.258450 0.326647 0.0315765 0.101741 0.876425  3.99902
 11:     7753.7758: 0.0230190 -0.257612 0.327650 0.0313323 0.101204 0.877826  3.99906
 12:     7753.6748: 0.0230090 -0.258208 0.327841 0.0293620 0.0994561 0.880746  3.99932
 13:     7753.6301: 0.0229765 -0.259765 0.328971 0.0288415 0.0978123 0.883304  4.00028
 14:     7753.5767: 0.0229281 -0.261830 0.330867 0.0289110 0.0965673 0.883349  4.00179
 15:     7753.5580: 0.0228758 -0.263897 0.332903 0.0291444 0.0962478 0.883663  4.00331
 16:     7753.4854: 0.0225334 -0.277016 0.345599 0.0274260 0.0950560 0.886480  4.01294
 17:     7753.4719: 0.0225323 -0.276695 0.345993 0.0279023 0.0940099 0.885988  4.01295
 18:     7753.4538: 0.0225135 -0.277276 0.346610 0.0282692 0.0941910 0.886249  4.01341
 19:     7753.4416: 0.0224933 -0.277896 0.347251 0.0281355 0.0941204 0.886129  4.01389
 20:     7753.4320: 0.0224549 -0.278505 0.349027 0.0282643 0.0937839 0.886357  4.01479
 21:     7753.3038: 0.0213907 -0.311227 0.381075 0.0282387 0.0939363 0.885763  4.03989
 22:     7753.2470: 0.0198191 -0.306779 0.375083 0.0267061 0.0894777 0.890618  4.05075
 23:     7753.1676: 0.0182480 -0.296072 0.364994 0.0269527 0.0904575 0.889971  4.05294
 24:     7753.1165: 0.0166676 -0.287184 0.356580 0.0268598 0.0911759 0.888953  4.04888
 25:     7753.0720: 0.0153164 -0.292789 0.362100 0.0272593 0.0919992 0.888716  4.01193
 26:     7753.0464: 0.0140359 -0.279643 0.349665 0.0276292 0.0924361 0.887756  4.04999
 27:     7753.0121: 0.0124586 -0.288551 0.358749 0.0278506 0.0928403 0.887006  4.04515
 28:     7753.0062: 0.0118041 -0.301139 0.371422 0.0277004 0.0928737 0.887052  4.04617
 29:     7753.0059: 0.0117413 -0.298655 0.368918 0.0278383 0.0929380 0.886806  4.04612
 30:     7753.0059: 0.0117547 -0.298461 0.368732 0.0277939 0.0929303 0.886877  4.04609
 31:     7753.0059: 0.0117695 -0.298774 0.369043 0.0277985 0.0929202 0.886878  4.04618
 32:     7753.0059: 0.0117636 -0.298690 0.368959 0.0277991 0.0929239 0.886874  4.04615

Final Estimate of the Negative LLH:
 LLH:  -18159.376    norm LLH:  -2.9546658 
             mu             ar1             ma1           omega          alpha1           beta1 
 1.73577740e-04 -2.98690367e-01  3.68959354e-01  6.05257206e-06  9.29239158e-02  8.86874472e-01 
          shape 
 4.04614910e+00 

R-optimhess Difference Approximated Hessian Matrix:
                    mu             ar1             ma1           omega          alpha1           beta1
mu     -29626609.54489   -9180.1325271   -5076.0398620  3.83282635e+08  4.36255687e+04  8.02938733e+04
ar1        -9180.13253   -7054.9901407   -7154.4458092  5.81740750e+06  6.97662687e+02  5.41664057e+02
ma1        -5076.03986   -7154.4458092   -7311.3552705  5.58039961e+06  6.38678510e+02  4.84689897e+02
omega  383282634.81092 5817407.5014527 5580399.6067520 -6.20511367e+12 -6.12244683e+08 -9.21076625e+08
alpha1     43625.56871     697.6626869     638.6785101 -6.12244683e+08 -9.56590330e+04 -1.17183443e+05
beta1      80293.87334     541.6640568     484.6898973 -9.21076625e+08 -1.17183443e+05 -1.60599870e+05
shape       2366.26467      37.4272552      35.3858391 -1.30460896e+07 -1.76644179e+03 -2.28726044e+03
                 shape
mu      2.36626467e+03
ar1     3.74272552e+01
ma1     3.53858391e+01
omega  -1.30460896e+07
alpha1 -1.76644179e+03
beta1  -2.28726044e+03
shape  -5.27928019e+01
attr(,"time")
Time difference of 0.194895983 secs

--- END OF TRACE ---


Time to Estimate Parameters:
 Time difference of 0.693755865 secs

Using formula(x) is deprecated when x is a character vector of length > 1.
  Consider formula(paste(x, collapse = " ")) instead.

Title:
 GARCH Modelling 

Call:
 garchFit(formula = ~arma(1, 1) + garch(1, 1), data = bmw, cond.dist = "std") 

Mean and Variance Equation:
 data ~ arma(1, 1) + garch(1, 1)
<environment: 0x7fe907230070>
 [data = bmw]

Conditional Distribution:
 std 

Coefficient(s):
         mu          ar1          ma1        omega       alpha1        beta1        shape  
 1.7358e-04  -2.9869e-01   3.6896e-01   6.0526e-06   9.2924e-02   8.8687e-01   4.0461e+00  

Std. Errors:
 based on Hessian 

Error Analysis:
         Estimate  Std. Error  t value Pr(>|t|)    
mu      1.736e-04   1.855e-04    0.936  0.34929    
ar1    -2.987e-01   1.370e-01   -2.180  0.02924 *  
ma1     3.690e-01   1.345e-01    2.743  0.00608 ** 
omega   6.053e-06   1.344e-06    4.502 6.72e-06 ***
alpha1  9.292e-02   1.312e-02    7.080 1.44e-12 ***
beta1   8.869e-01   1.542e-02   57.528  < 2e-16 ***
shape   4.046e+00   2.315e-01   17.481  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Log Likelihood:
 18159.376    normalized:  2.9546658 

Description:
 Sun Aug  1 10:22:10 2021 by user:  

ma2 <- arima.sim(n = 1000, list(ma = c(-0.2279, 0.2488)))
par(mfrow=c(2,1))
plot(ma2)
acf(ma2)

ar1 <- arima.sim(n = 1000, list(ar = c(0.9))) 
par(mfrow=c(2,1)) 
plot(ar1) 
acf(ar1)

ar5 <- arima.sim(n = 1000, list(ar = c(-0.2279, 0.2488, -0.5, 0.2, 0.6)))
par(mfrow=c(2,1))
plot(ar5)
acf(ar5)

library(prophet) 
library(TSstudio)

model <- tbats(elecequip, use.box.cox = TRUE, use.damped.trend = F,start.p=2,max.p=2,start.q=1,max.q=1)
plot(forecast(model, h=12))

library(fpp2)

── Attaching packages ────────────────────────────────────────────────────────────────────── fpp2 2.4 ──
✓ ggplot2   3.3.3     ✓ expsmooth 2.3  
✓ fma       2.4       

plot(elecequip)

data_points = 1000 
X <- cumsum(rnorm(data_points, 0, 1)) 
Y <- 2*X + 10 + rnorm(data_points, 0, 1)

par(mfrow=c(2,2))
plot(X, main="Time Series X")
plot(diff(X), main="Time Series of First Differences of X") 

plot(Y, main="Time Series Y") 
plot(diff(Y), main="Time Series of First Differences of Y")

adf.test(X, k=1)

    Augmented Dickey-Fuller Test

data:  X
Dickey-Fuller = -2.304394, Lag order = 1, p-value = 0.44946
alternative hypothesis: stationary

adf.test(diff(X), k=0)

p-value smaller than printed p-value

    Augmented Dickey-Fuller Test

data:  diff(X)
Dickey-Fuller = -30.42674, Lag order = 0, p-value = 0.01
alternative hypothesis: stationary

adf.test(Y, k=1)

    Augmented Dickey-Fuller Test

data:  Y
Dickey-Fuller = -2.33245, Lag order = 1, p-value = 0.437584
alternative hypothesis: stationary

adf.test(diff(Y), k=0)

p-value smaller than printed p-value

    Augmented Dickey-Fuller Test

data:  diff(Y)
Dickey-Fuller = -35.79313, Lag order = 0, p-value = 0.01
alternative hypothesis: stationary

lm(Y~X)

Call:
lm(formula = Y ~ X)

Coefficients:
(Intercept)            X  
    9.99348      2.00063  

spread = Y - 10.0009 - 1.997X

Error: unexpected symbol in "spread = Y - 10.0009 - 1.997X"

library(ggplot2) 
data(economics) 
lmMod <- lm(pce ~ pop, data=economics) 
lmMod

Call:
lm(formula = pce ~ pop, data = economics)

Coefficients:
 (Intercept)           pop  
-1.97965e+04   9.57251e-02  

acf(lmMod$residuals) # highly autocorrelated from the picture.

lmtest::dwtest(lmMod)

    Durbin-Watson test

data:  lmMod
DW = 0.002159001, p-value < 2.22e-16
alternative hypothesis: true autocorrelation is greater than 0

scatter.smooth(x=cars$speed, y=cars$dist, main="Dist ~ Speed")

linearMod <- lm(dist ~ speed, data=cars)
linearMod

Call:
lm(formula = dist ~ speed, data = cars)

Coefficients:
(Intercept)        speed  
  -17.57909      3.93241  

# Create Training and Test data - 
set.seed(100) # setting seed to reproduce results of random sampling 
trainingRowIndex <- sample(1:nrow(cars), 0.8*nrow(cars)) # row indices for training data 
trainingData <- cars[trainingRowIndex, ] # model training data 
testData <- cars[-trainingRowIndex, ] # test data

# Build the model on training data - 
lmMod <- lm(dist ~ speed, data=trainingData) # build the model 
distPred <- predict(lmMod, testData) # predict distance 
actuals_preds <- data.frame(cbind(actuals=testData$dist, predicteds=distPred)) # make actuals_predicteds dataframe 

mean((actuals_preds$predicteds - actuals_preds$actuals)^2)

[1] 267.000242

library("ggplot2") 
library("tidyverse")

data(cars)
head(cars)

plot(cars)

summary(cars)

     speed           dist       
 Min.   : 4.0   Min.   :  2.00  
 1st Qu.:12.0   1st Qu.: 26.00  
 Median :15.0   Median : 36.00  
 Mean   :15.4   Mean   : 42.98  
 3rd Qu.:19.0   3rd Qu.: 56.00  
 Max.   :25.0   Max.   :120.00  

boxplot(cars)

cars.loess <- select(cars, speed, dist)
cars.Train <- cars.loess[1:35,] 
cars.Test <- cars.loess[36:50,]

cars.LoessModel <- loess( dist ~ speed, cars.Train, control = loess.control(surface = "direct") )
cars.LoessModel

Call:
loess(formula = dist ~ speed, data = cars.Train, control = loess.control(surface = "direct"))

Number of Observations: 35 
Equivalent Number of Parameters: 4.82 
Residual Standard Error: 13.8708 

forwardSeq <- as.vector(cars.Test$speed)
forwardSeq

 [1] 19 19 19 20 20 20 20 20 22 23 24 24 24 24 25

cars.Predicted <- predict(cars.LoessModel, data.frame(speed=forwardSeq), se=TRUE )
cars.Predicted

$fit
          1           2           3           4           5           6           7           8           9 
 79.7288965  79.7288965  79.7288965 106.5018421 106.5018421 106.5018421 106.5018421 106.5018421 183.8314219 
         10          11          12          13          14          15 
234.6318396 293.6078199 293.6078199 293.6078199 293.6078199 360.7953680 

$se.fit
          1           2           3           4           5           6           7           8           9 
 12.4673442  12.4673442  12.4673442  21.9702630  21.9702630  21.9702630  21.9702630  21.9702630  50.5278805 
         10          11          12          13          14          15 
 69.6316220  92.0038632  92.0038632  92.0038632  92.0038632 117.6754780 

$residual.scale
[1] 13.8707798

$df
[1] 29.5547794

cars.Test$predicted_dist <- cars.Predicted$fit
cars.Test

cars.TestPlot <- select(cars.Test, speed, predicted_dist) 
cars.TestPlot <- rename(cars.TestPlot, dist = predicted_dist)

ggplot(cars.Train, aes(x = speed, y = dist)) + geom_point() + 
  geom_smooth(stat="smooth", position = "identity", method = "loess", formula = y ~ x, se = TRUE, na.rm = FALSE, inherit.aes = TRUE)+ geom_line(data = cars.TestPlot, aes(y=dist, x=speed, colour="#000099"), show.legend = FALSE, size = 2)

ggplot(cars.loess, aes(x = speed, y = dist)) + geom_point() + geom_smooth( stat="smooth", position = "identity", method = "loess", formula = y ~ x, se = TRUE, na.rm = FALSE, inherit.aes = TRUE ) + geom_line(data = cars.TestPlot, aes(y=dist, x=speed, colour="#000099"), show.legend = FALSE, size = 2)

## 70% of the sample size
smp_size <- floor(0.70 * nrow(cars.loess))
## set the seed to make your partition reproducible 
set.seed(123) 
train_ind <- sample(seq_len(nrow(cars.loess)), size = smp_size) 
cars.Train <- cars.loess[train_ind, ] 
cars.Test <- cars.loess[-train_ind, ]

cars.LoessModel <- loess(dist ~ speed, cars.Train, control = loess.control(surface = "direct"))
forwardSeq <- as.vector(cars.Test$speed)
cars.Predicted <- predict(cars.LoessModel, data.frame(speed=forwardSeq), se=TRUE) 
cars.Test$predicted_dist <- cars.Predicted$fit 
cars.TestPlot <- select(cars.Test, speed, predicted_dist) 
cars.TestPlot <- rename(cars.TestPlot, dist = predicted_dist)

ggplot(cars.loess, aes(x = speed, y = dist)) + geom_point() + geom_smooth( stat="smooth", position = "identity", method = "loess", formula = y ~ x, se = TRUE, na.rm = FALSE, inherit.aes = TRUE ) + geom_line(data = cars.TestPlot, aes(y=dist, x=speed, colour="#000099"), show.legend = FALSE, size = 2)

library(fpp2)
plot(elecequip)

library(forecast)
par(mfrow=c(2,1))

model <- naive(elecequip)
plot(predict(model, n.ahead=24))
model <- snaive(elecequip)
plot(predict(model, n.ahead=24))

plot(stl(elecequip, t.window=13, s.window="periodic", robust=TRUE))
par(mfrow=c(1,1))

model <- stl(elecequip, t.window=13, s.window="periodic", robust=TRUE)
plot(forecast(model, method = "naive", h=12))

plot(decompose(elecequip))

model <- ets(elecequip) 
plot(forecast(model,h=12))

model <- auto.arima(elecequip) 
model

Series: elecequip 
ARIMA(4,0,1)(0,1,1)[12] 

Coefficients:
           ar1        ar2       ar3        ar4        ma1       sma1
      1.162498  -0.029398  0.223521  -0.383994  -0.595393  -0.810746
s.e.  0.146681   0.140929  0.120976   0.070799   0.156950   0.078211

sigma^2 estimated as 10.9033:  log likelihood=-482.97
AIC=979.93   AICc=980.57   BIC=1002.4

plot(forecast(model,h=12)) 

forecast(arima(elecequip, order=c(4,0,1), seasonal = c(0,1,1)), h=12)

NA

model <- tbats(elecequip, use.box.cox = TRUE, use.damped.trend = F,start.p=2,max.p=2,start.q=1,max.q=1)
plot(forecast(model, h=12))

library(prophet) 
library(TSstudio)
df = ts_to_prophet(elecequip) 
next_time_step = function(y, is_end = F){ 
  if(is_end){ new_start = y }
  else{ new_start = end(y) } 
  if(new_start[2]==12){ 
    new_start[1]=new_start[1]+1 new_start[2]=1 }else{ new_start[2]=new_start[2]+1 } return(new_start)}